The six operations for sheaves on Artin stacks I: Finite coefficients

FINITE COEFFICIENTS

by YVES LASZLO and MARTINOLSSON

ABSTRACT

In this paper we develop a theory of Grothendieck’s six operations of lisse-étale constructible sheaves on Artin stacks locally of finite type over certain excellent schemes of finite Krull dimension. We also give generalizations of the classical base change theorems and Kunneth formula to stacks, and prove new results about cohomological descent for unbounded complexes.

1. Introduction

We denote by Λ a Gorenstein local ring of dimension 0 and characteristic l.

Let S be an affine excellent finite-dimensional scheme and assume l is invertible on S. We assume that all S-schemes of finite type X satisfy cdl(X)<∞ (see 1.0.1 for more discussion of this). For an algebraic stack X locally of finite type over S and ∗ ∈ {+,−,b,∅,[a,b]} we write D^∗_c(X) for the full subcategory of the derived category D^∗(X) of complexes of Λ-modules on the lisse-étale site of X with constructible cohomology sheaves. We will also consider the variant subcategories D^(∗)_c (X) ⊂ Dc(X) consisting of complexes K such that for any quasi-compact open U ⊂X the restriction K|U is in D^∗_c(U).

In this paper we develop a theory of Grothendieck’s six operations of lisse- étale constructible sheaves on Artin stacks locally of finite type over S¹. In forth- coming papers, we will also develop a theory of adic sheaves and perverse sheaves for Artin stacks. In addition to being of basic foundational interest, we hope that the development of these six operations for stacks will have a number of applica- tions. Already the work done in this paper (and the forthcoming ones) provides the necessary tools needed in several papers on the geometric Langland’s program (e.g.

[21], [19], [7]). We hope that it will also shed further light on the Lefschetz trace formula for stacks proven by Behrend [1], and also to versions of such a formula for stacks not necessarily of finite type. We should also remark that recent work of Toen should provide another approach to defining the six operations for stacks, and in fact should generalize to a theory for n-stacks.

Partially supported by NSF grant DMS-0714086 and an Alfred P. Sloan Research Fellowship

1 In fact our method could apply to other situations like analytic stacks or non separated analytic varieties.

DOI 10.1007/s10240-008-0011-6

Let us describe more precisely the contents of this papers. For a finite type morphism f :X →Y of stacks locally of finite type over S we define functors

Rf_∗:D⁽⁺⁾_c (X)→D⁽⁺⁾_c (Y ), Rf_!:D⁽⁻⁾_c (X)→D⁽⁻⁾_c (Y ), Lf^∗:Dc(Y )→Dc(X), Rf^! :Dc(Y )→Dc(X), Rhom:D⁽⁻⁾_c (X)^op×D⁽⁺⁾_c (X)→D⁽⁺⁾_c (X),

and

(−)⊗(−)^L :D⁽⁻⁾_c (X)×D⁽⁻⁾_c (X)→D⁽⁻⁾_c (X)

satisfying all the usual adjointness properties that one has in the theory for schemes².

The main tool is to define f_!,f^!, even for unbounded constructible complexes, by duality. One of the key points is that, as observed by Laumon, the dualizing complex is a local object of the derived category and hence has to exist for stacks by glueing (see 2.3.3). Notice that this formalism applies to non-separated schemes, giving a theory of cohomology with compact supports in this case. Previ- ously, Laumon and Moret-Bailly constructed the truncations of dualizing complexes for Bernstein–Lunts stacks (see [20]). Our constructions reduces to theirs in this case. Another approach using a dual version of cohomological descent has been suggested by Gabber but seems to be technically much more complicated.

1.0.1. Remark. — The cohomological dimension hypothesis on schemes of finite type over S is achieved for instance if S is the spectrum of a finite field or of a separably closed field. In dimension 1, it will be achieved for instance for the spectrum of a complete discrete valuation field with residue field either finite or separably closed, or if S is a smooth curve over C,Fq (cf. [13, exp. X] and [26]).

In these situations, cdl(X) is bounded by a function of the dimension dim(X). 1.1. Conventions. — In order to develop the theory over an excellent base S as above, we use the recent finiteness results of Gabber [9] and [10]. A complete account of these results will soon appear in a writeup of the seminar on Gabber’s work [15]. However, the reader uncomfortable with this theory may assume that S is an affine regular, noetherian scheme of dimension ≤1.

Our conventions about stacks are those of [20].

Let X be an algebraic stack locally of finite type over S and let K∈D⁺_c (X) be a complex of Λ-modules. Following [14, I.1.1], we say that K has finite quasi- injective dimension if there exists an integer n such that for any constructible sheaf of Λ-modules F we have

Extⁱ(F,K)=0, for i > n.

2 We will often writef^∗,f^!,f_∗,f_! forLf^∗,Rf^!,Rf_∗,Rf_!.

For K∈D⁽⁺⁾_c (X) we say that K has locally finite quasi-injective dimension if for every quasi-compact open substack U _→ X the restriction K|U ∈ D⁺_c(U) has finite quasi-injective dimension.

In this paper we work systematically with unbounded complexes. The theory of derived functors for unbounded complexes on topological spaces is due to Spal- tenstein [27], and for Grothendieck categories Serpé [25]. An excellent reference for unbounded homological algebra is the book of Kashiwara and Schapira [16].

Recall that for a ringed topos (T,OT) one has functors [16, Theorem 18.6.4], (−)⊗^L (−):D(OT)×D(OT)→D(OT)

and

Rhom(−,−):D(OT)×D(OT)^op →D(OT),

and for a morphism f :(T,OT)→(S,OS) of ringed topos one has functors [16, Theorem 18.6.9] and the line preceding this theorem,

Rf_∗ :D(OT)→D(OT) and

Lf^∗ :D(OS)→ D(OT).

Moreover these satisfy the usual adjunction properties that one would expect from the theory for the bounded derived category.

All the stacks we will consider will be locally of finite type over S. As in [20, lemme 12.1.2], the lisse-étale topos Xlis-ét can be defined using the site Lisse-Et(X) whose objects are S-morphisms u:U→X where U is an algebraic space which is separated and of finite type over S. The topology is generated by the pretopology such that the covering families are finite families (Ui,ui)→(U,u) such that

Ui→U is surjective and étale (use the comparison theorem [13, III.4.1] remembering X is locally of finite type over S). Notice that products over X are representable in Lisse-Et(X), simply because the diagonal morphism X →X×SX is representable and separated by definition [20].

If C is a complex of sheaves and d a locally constant valued function C(d) is the Tate twist and C[d] the shifted complex. We denote C(d)[2d] by Cd . We fix once and for all a dualizing complex Ω^S on S. In the case when S is regular of dimension 0 or 1 we take Ω^S =Λdim(S) [4, “Dualité”].

2. Homological algebra

2.1. Existence of K-injectives. — Let (S,O) denote a ringed site, and let C denote a full subcategory of the category of O-modules on S. Let M be

a complex of O-modules on S. By [27, 3.7], there exists a morphism of com- plexes f :M→I with the following properties:

(i) I=lim←−In where each In is a bounded below complex of flasque O-mod- ules.

(ii) The morphism f is induced by a compatible collection of quasi-iso- morphisms fn :τ≥−nM→In.

(iii) For every n the map In → In−1 is surjective with kernel Kn a bounded below complex of flasque O-modules.

(iv) For any pair of integers n and i the sequence 0→ Kⁱ_n →Iⁱ_n→Iⁱ_n−1 →0

(2.1.i)

is split.

2.1.1. Remark. — In fact [27, 3.7], shows that we can choose In and Kn to be complexes of injective O-modules (in which case (iv) follows from (iii)). However, for technical reasons it is sometimes useful to know that one can work just with flasque sheaves.

We make the following finiteness assumption, which is the analog of [27, 3.12 (1)].

2.1.2. Assumption. — For any object U ∈ S there exists a covering {Ui → U}ⁱ∈I

and an integer n0 such that for any sheaf of O-modules F∈C we have Hⁿ(Ui,F)=0 for all n≥n0.

2.1.3. Example. — Let S = Lisse-Et(X) be the lisse-étale site of an alge- braic S-stack locally of finite type X and O a constant local Gorenstein ring of dimension 0 and characteristic invertible on S. Then the class C of all O-sheaves, cartesian or not, satisfies the assumption. Indeed, if U∈ S is of finite type over S and F ∈S, one has Hⁿ(U,F)=Hⁿ(Uét,FU)³ which is zero for n bigger than a constant depending only on U (and not on F). Therefore, one can take the trivial covering in this case. We could also take O =OX and C to be the class of quasi-coherent sheaves.

With Hypothesis 2.1.2, one has the following criterion for f being a quasi- isomorphism (cf. [27, 3.13]).

2.1.4. Proposition. — Assume that H^j(M) ∈ C for all j. Then the map f is a quasi-isomorphism. In particular, if each In is a complex of injective O-modules then by [27, 2.5], f :M→I is a K-injective resolution of M.

3 Cf. 3.3.1 below

Proof. — For a fixed integer j, the map H^j(M)→H^j(In) is an isomorphism for n sufficiently big. Since this isomorphism factors as

H^j(M)→H^j(I)→H^j(In) (2.1.ii)

it follows that the map H^j(M)→H^j(I) is injective.

To see that H^j(M) → H^j(I) is surjective, let U ∈ S be an object and γ ∈Γ(U,I^j) an element with dγ =0 defining a class in H^j(I)(U). Since I=lim←−In

the class γ is given by a compatible collection of sections γⁿ ∈ Γ(U,I_n^j) with dγⁿ=0.

Let (U = {Ui→U},n0) be the data provided by 2.1.2. Let N be an integer greater than n0−j. For m≥N and Ui ∈U the sequence

Γ

Ui,K^j−1_m

→Γ

Ui,K^j_m

→Γ

Ui,K^j+1_m

→Γ

Ui,K^j+2_m (2.1.iii)

is exact. Indeed Km is a bounded below complex of flasque sheaves quasi-iso- morphic to H^−m(M)[m], and therefore the exactness is equivalent to the statement that the groups

H^j(Ui,H^−m(M)[m])=H^j+m(Ui,H^−m(M))

and

H^j+1(Ui,H^−m(M)[m])=H^j+m+1(Ui,H^−m(M))

are zero. This follows from the assumptions and the observation that j+m≥j+N > j+n0−j =n0,

j+1+m≥j+1+N≥j+n0−j =n0.

Since the maps Γ(Ui,I^r_m)→ Γ(Ui,I^r_m−1) are also surjective for all m and r, it follows from [27, 0.11], applied to the system

Γ

Ui,I_m^j−1

→Γ Ui,Iⁱ_m

→ Γ

Ui,I_m^j+1

→Γ

Ui,I_m^j+2 (2.1.iv)

that the map

H^j(Γ(Ui,I))→H^j(Γ(Ui,Im)) (2.1.v)

is an isomorphism.

Then since the map H^j(M)→H^j(Im) is an isomorphism it follows that for every i the restriction of γ to Ui is in the image of H^j(M)(Ui). Next consider a fibred topos T →D with corresponding total topos T_• [13, VI.7]. We call T_• a D-simplicial topos. Concretely, this means that for each i ∈D the fiber Ti is a topos and that any δ∈HomD(i,j) comes together with a morphism of topos δ : Ti→Tj such that δ⁻¹ is the inverse image functor of the fibred structure. The objects of the total topos are simply collections (Fi∈Ti)i∈D together with functorial transition morphisms δ⁻¹Fj→Fi for any δ∈HomD(i,j). We assume furthermore that T• is ringed by a O• and that for any δ ∈ HomD(i,j), the morphism δ:(Ti,Oi)→(Tj,Oj) is flat.

2.1.5. Example. — Let ∆⁺ be the category whose objects are the ordered sets [n] = {0, ...,n} (n ∈ N) and whose morphisms are injective order-preserving maps. Let D be the opposite category of ∆⁺. In this case T_. is called a strict simplicial topos. For instance, if U→X is a presentation, the simplicial algebraic space U_• = cosq₀(U/X) defines a strict simplicial topos U_•lis-ét whose fiber over [n] is Unlis-ét. For a morphism δ: [n] → [m] in ∆⁺ the morphism δ: Tm→Tn is induced by the (smooth) projection Um→Un defined by δ∈Hom_∆^+opp([m],[n]).

2.1.6. Example. — Let N be the natural numbers viewed as a category in which Hom(n,m) is empty unless m ≥ n in which case it consists of a unique element. For a topos T we can then define an N-simplicial topos T^N. The fiber over n of T^N is T and the transition morphisms by the identity of T. The topos T^N is the category of projective systems in T. If O• is a constant projective sys- tem of rings then the flatness assumption is also satisfied, or more generally if δ⁻¹On → Om is an isomorphism for any morphism δ : m → n in N then the flatness assumption holds.

Let C_• be a full subcategory of the category of O_•-modules on a ringed D-simplicial topos (T_•,O_•). For i ∈ D, let ei : Tn→T_• the morphism of topos defined by e⁻_i ¹F_• = Fn (cf. [13, V^bis.1.2.11]). Recall that the family e⁻_i ¹,i ∈ D is conservative. Let Ci denote the essential image of C_• under e_i⁻¹ (which coincides with e_i^∗ on Mod(T•,O_•) because e_i⁻¹O_•=Oi).

2.1.7. Assumption. — For every i∈D the ringed topos (Ti,Oi) is isomorphic to the topos of a ringed site satisfying 2.1.2 with respect to Ci.

2.1.8. Example. — Let T• be the topos (Xlis-ét)^N of a S-stack locally of finite type. Then, the full subcategory C• of Mod(T•,O•) whose objects are families Fi

of cartesian modules satisfies the hypothesis.

Let M be a complex of O_•-modules on T_•. Again by [27, 3.7], there exists a morphism of complexes f :M→I with the following properties:

(S i) I = lim←−In where each In is a bounded below complex of injective modules.

(S ii) The morphism f is induced by a compatible collection of quasi-iso- morphisms fn :τ≥−nM→In.

(S iii) For every n the map In→In−1 is surjective with kernel Kn a bounded below complex of injective O-modules.

(S iv) For any pair of integers n and i the sequence 0→Kⁱ_n→Iⁱ_n →Iⁱ_n−1→0

(2.1.vi)

is split.

2.1.9. Proposition. — Assume that H^j(M)∈C• for all j. Then the morphism f is a quasi-isomorphism and f :M→I is a K-injective resolution of M.

Proof. — By [27, 2.5], it suffices to show that f is a quasi-isomorphism. For this in turn it suffices to show that for every i∈D the restriction e^∗_if :e^∗_iM→e_i^∗I is a quasi-isomorphism of complexes of Oi-modules since the family e^∗_i = e⁻_i ¹ is conservative. But e^∗_i :Mod(T•,O_•)→Mod(Ti,Oi) has a left adjoint ei! defined by

[ei!(F)]j =

δ∈HomD(j,i)

δ^∗F

with the obvious transition morphisms. It is exact by the flatness of the morph- isms δ. It follows that e^∗_i takes injectives to injectives and commutes with direct limits. We can therefore apply 2.1.4 to e^∗_iM→e^∗_iI to deduce that this map is

a quasi-isomorphism.

In what follows we call a K-injective resolution f : M → I obtained from data (i)–(iv) as above a Spaltenstein resolution.

The main technical lemma is the following.

2.1.10. Lemma. — Let :(T•,O•)→(S,Ψ) be a morphism of ringed topos, and let C be a complex of O_•-modules. Assume that

1.Hⁿ(C)∈C• for all n.

2. There exists i0 such that Rⁱ∗Hⁿ(C)=0 for any n and any i > i0. Then, if j≥ −n+i0, we have R^j∗C=R^j∗τ≥−nC.

Proof. — By 2.1.9 and assumption (1), there exists a Spaltenstein resolution f : C → I of C. Let Jn := ∗In and Fn := ∗Kn. Since the sequences (2.1.vi) are

split, the sequences

0→ Fn →Jn→J_n−1 →0 (2.1.vii)

are exact.

The exact sequence (2.1.vi) and property (S ii) defines a distinguished triangle Kn →τ≥−nC→τ≥−n+1C

showing that Kn is quasi-isomorphic to H⁻ⁿ(C)[n]. Because Kn is a bounded below complex of injectives, one gets

R∗H⁻ⁿ(C)[n] =∗Kn

and accordingly

R^j+n∗H⁻ⁿ(C)=H^j(∗Kn)=H^j(Fn).

By assumption (2), we have therefore H^j(Fn)=0 for j >−n+i0. By [27, 0.11], this implies that

H^j(lim←−Jn)→H^j(Jn)

is an isomorphism for j ≥ −n+i0. But, by adjunction, ∗ commutes with projective limit. In particular, one has

lim←−Jn =∗I, and by (S i) and (S ii)

R∗C=∗I and R∗τ≥−nC=∗Jn. Thus for any n such that j ≥ −n+i0 one has

R^j∗C=H^j(∗I)=H^j(Jn)=R^j∗τ≥−nC. (2.1.viii)

2.1.11. Remark. — An important special case of Lemma 2.1.10 is the fol- lowing. Take D to be the category with one element and one morphism so that :(T,O)→(S,Ψ) is just a morphism of ringed topos. Let C be a full subcate- gory of the category of O-modules such that (T,O) is isomorphic to the ringed topos associated to a ringed site satisfying Assumption 2.1.2 with respect to C. Let C be a complex of O-modules such that

1.Hⁿ(C)∈C for all n.

2. There exists an integer i0 such that Rⁱ∗Hⁿ(C)=0 for any n and i > i0. Then by Lemma 2.1.10 the natural map R^j∗C→ R^j∗τ≥−nC is an isomorphism for j ≥ −n+i0.

2.2. The descent theorem. — Let (T•,O_•) be a simplicial or strictly simplicial⁴ ringed topos (D=∆^opp or D=∆^+opp), let (S,Ψ) be another ringed topos, and let :(T•,O_•)→(S,Ψ) be an augmentation. Assume that is a flat morphism (i.e.

for every i ∈D, the morphism of ringed topos (Ti,Oi)→(S,Ψ) is a flat morph- ism), and furthermore that for every morphism δ :i → j in D the corresponding morphism of ringed topos (Ti,Oi)→(Tj,Oj) is flat.

Let C be a full subcategory of the category of Ψ-modules, and assume that C is closed under kernels, cokernels and extensions (one says that C is a Serre subcategory). Let D(S) denote the derived category of Ψ-modules, and let D_C(S)⊂ D(S) be the full subcategory consisting of complexes whose cohomology sheaves are in C. Let C• denote the essential image of C under the functor ^∗ :Mod(Ψ)→ Mod(O_•).

We assume the following condition holds:

2.2.1. Assumption. — Assumption 2.1.7 holds (with respect to C•), and ^∗ :C→C•

is an equivalence of categories with quasi-inverse R∗.

2.2.2. Lemma. — The full subcategory C_• ⊂ Mod(O•) is closed under extensions, kernels and cokernels.

Proof. — Consider an extension of sheaves of O_•-modules 0 −−−→ ^∗F1 −−−→ E −−−→ ^∗F2 −−−→ 0, (2.2.i)

where F1,F2 ∈ C. Since R¹∗^∗F1 = 0 and the maps Fi → R⁰∗^∗Fi are isomor- phisms, we obtain by applying ^∗∗ a commutative diagram with exact rows

0 −−−→ ^∗F1 −−−→ ^∗∗E −−−→ ^∗F2 −−−→ 0

id⏐⏐ ^α⏐⏐ ⏐⏐^id

0 −−−→ ^∗F1 −−−→ E −−−→ ^∗F2 −−−→ 0. (2.2.ii)

It follows that α is an isomorphism. Furthermore, since C is closed under ex- tensions we have ∗E ∈ C. Let f ∈ Hom(^∗F1, ^∗F2). There exists a unique ϕ ∈Hom(F1,F2) such that f =^∗ϕ. Because ^∗ is exact, it maps the kernel and cokernel of ϕ, which are objects of C, to the kernel and cokernel of f respectively.

Therefore, the latter are objects of C_•.

Let D(T•) denote the derived category of O•-modules, and let D_C•(T•) ⊂ D(T•) denote the full subcategory of complexes whose cohomology sheaves are in C_•.

4 One could replace simplicial by multisimplicial

Since is a flat morphism, we obtain a morphism of triangulated categories (the fact that these categories are triangulated comes precisely from the fact that both C and C_• are Serre categories [11]).

^∗:D_C(S)→D_C•(T•).

(2.2.iii)

2.2.3. Theorem. — The functor ^∗ of (2.2.iii) is an equivalence of triangulated cate- gories with quasi-inverse given by R∗.

Proof. — Note first that if M_• ∈ D_C•(T•), then by Lemma 2.1.10, for any integer j there exists n0 such that R^j∗M_• =R^j∗τ≥n0M_•. In particular, we get by induction R^j∗M_•∈C. Thus R∗ defines a functor

R∗ :D_C•(T•)→D_C(S).

(2.2.iv)

To prove 2.2.3 it suffices to show that for M_• ∈ D_C•(T_•) and F ∈ D_C(S) the adjunction maps

^∗R∗M_•→M_•, F→R∗^∗F (2.2.v)

are isomorphisms. For this note that for any integers j and n there are commutative diagrams

^∗R^j∗M_• −−−→ H^j(M_•)

⏐⏐ ⏐⏐

^∗R^j∗τ≥nM_• −−−→ H^j(τ≥nM_•), (2.2.vi)

and

H^j(F) −−−→ R^j∗^∗F

⏐⏐ ⏐⏐

H^j(τ≥nF) −−−→ R^j∗^∗τ≥nF. (2.2.vii)

By the observation at the begining of the proof, there exists an integer n so that the vertical arrows in the above diagrams are isomorphisms. This reduces the proof 2.2.3 to the case of a bounded below complex. In this case one reduces by devissage to the case when M_•∈C• and F∈C in which case the result holds by

assumption.

The theorem applies in particular to the following examples.

2.2.4. Example. — Let S be an algebraic space and X_• → S a flat hyper- cover by algebraic spaces. Let X⁺_• → S denote the associated strictly simplicial space with S-augmentation. We then obtain an augmented strictly simplicial topos :(X⁺_•,ét,OX⁺_•,ét)→(Sét,Oét). Note that this augmentation is flat. Let C denote the category of quasi-coherent sheaves on Sét. Then the category C_• is the category of cartesian sheaves of OX⁺_•,ét-modules whose restriction to each Xn is quasi-coherent.

Let Dqcoh(X⁺_•) denote the full subcategory of the derived category of OX⁺_•,ét-modules whose cohomology sheaves are quasi-coherent, and let Dqcoh(S) denote the full sub- category of the derived category of OS_ét-modules whose cohomology sheaves are quasi-coherent. Theorem 2.2.3 then shows that the pullback functor

^∗ :D_qcoh(S)→D_qcoh(X⁺_•) (2.2.viii)

is an equivalence of triangulated categories with quasi-inverse R∗.

2.2.5. Example. — Let X be an algebraic stack and let U_• → X be a smooth hypercover by algebraic spaces. Let D(X) denote the derived category of sheaves of OXlis-ét-modules in the topos Xlis-ét, and let Dqcoh(X) ⊂ D(X) be the full subcategory of complexes with quasi-coherent cohomology sheaves.

Let U⁺_• denote the strictly simplicial algebraic space obtained from U_• by forgetting the degeneracies. Since the Lisse-Étale topos is functorial with respect to smooth morphisms, we therefore obtain a strictly simplicial topos U⁺_•lis-ét and a flat morphism of ringed topos

:

U⁺_•lis-ét,OU⁺_•lis-ét

→(X^lis-ét,OXlis-ét).

Then 2.2.1 holds with C equal to the category of quasi-coherent sheaves on X. The category C_• in this case is the category of cartesian OU⁺_•lis-ét-modules M_• such that the restriction Mn is a quasi-coherent sheaf on Un for all n. By 2.2.3 we then obtain an equivalence of triangulated categories

Dqcoh(X)→Dqcoh

U⁺_•,lis-ét , (2.2.ix)

where the right side denotes the full subcategory of the derived category of OU⁺_•lis-ét-modules with cohomology sheaves in C_•.

On the other hand, there is also a natural morphism of ringed topos π :

U⁺_•lis-ét,OU⁺_•lis-ét

→

U⁺_•ét,OU⁺_•ét

with π∗ and π^∗ both exact functors. Let Dqcoh(U⁺_•ét) denote the full subcate- gory of the derived category of OU⁺_•ét-modules consisting of complexes whose co- homology sheaves are quasi-coherent (i.e. cartesian and restrict to a quasi-coherent sheaf on each Unét). Then π induces an equivalence of triangulated categories

Dqcoh(U⁺_•ét) Dqcoh(U⁺_•lis-ét). Putting it all together we obtain an equivalence of

triangulated categories Dqcoh(X^lis-ét)Dqcoh(U⁺_•ét).

2.2.6. Example. — Let X be an algebraic stack locally of finite type over S and O be a constant local Gorenstein ring of dimension 0 and of character- istic invertible on S. Let U_• → X be a smooth hypercover by algebraic spaces, and T_• the localized topos Xlis-ét|U_•. Take C to be the category of constructible sheaves of O-modules. Then 2.2.3 gives an equivalence D_c(Xlis-ét) D_c(T•,Λ). On the other hand, there is a natural morphism of topos λ : T_• → U_•,ét and one sees immediately that λ∗ and λ^∗ induce an equivalence of derived categories D_c(T•,Λ)D_c(U_•,ét,Λ). It follows that D_c(Xlis-ét)D_c(U_•,ét).

2.3. The BBD gluing lemma. — The purpose of this section is to explain how to modify the proof of the gluing lemma [2, 3.2.4], for unbounded complexes.

Let ∆ denote the strictly simplicial category of finite ordered sets with in- jective order preserving maps, and let ∆⁺ ⊂ ∆ denote the full subcategory of nonempty finite ordered sets. For a morphism α in ∆ we write s(α) (resp. b(α)) for its source (resp. target).

Let T be a topos and U_· → e a strictly simplicial hypercovering of the initial object e ∈ T. For [n] ∈ ∆ write Un for the localized topos T|U_n where by definition we set U_∅ = T. Then we obtain a strictly simplicial topos U_· with an augmentation π :U_· →T.

Let Λ be a sheaf of rings in T and write also Λ for the induced sheaf of rings in U_· so that π is a morphism of ringed topos.

Let C_· denote a full substack of the fibered and cofibered category over ∆ [n] →(category of sheaves of Λ-modules in Un)

such that each Cn is a Serre subcategory of the category of Λ-modules in Un. For any [n] we can then form the derived category D_C(Un,Λ) of complexes of Λ-modules whose cohomology sheaves are in Cn. The categories D_C(Un,Λ) form a fibered and cofibered category over ∆.

We make the following assumptions on C:

2.3.1. Assumption. — (i) For any [n] the topos Un is equivalent to the topos associated to a site Sn such that for any object V ∈ Sn there exists an integer n0 and a covering {Vj → V} in Sn such that for any F∈Cn we have Hⁿ(Vj,F)=0 for all n≥n0.

(ii) The natural functor

C_∅→(cartesian sections of C|∆⁺ over ∆⁺) is an equivalence of categories.

(iii) The category D(T,Λ) is compactly generated.

2.3.2. Remark. — The case we have in mind is when T is the lisse-étale topos of an algebraic stack X locally of finite type over an affine regular,

noetherian scheme of dimension ≤ 1, U_· is given by a hypercovering of X by schemes, Λ is a Gorenstein local ring of dimension 0 and characteristic l invert- ible on X, and C is the category of constructible Λ-modules. In this case the category Dc(Xlis-et,Λ) is compactly generated. Indeed a set of generators is given by sheaves j_!Λ[i] for i∈Z and j :U→ X an object of the lisse-étale site of X.

There is also a natural functor

D_C(T,Λ)→(cartesian sections of [n] →D_C(Un,Λ) over ∆⁺).

(2.3.i)

2.3.3. Theorem. — Let [n] → Kn ∈ D_C(Un,Λ) be a cartesian section of [n] → D_C(Un,Λ) over ∆⁺ such that Extⁱ(K0,K0)= 0 for all i < 0. Then (Kn) is induced by a unique object K∈D_C(T,Λ) via the functor (2.3.i).

The uniqueness is the easy part:

2.3.4. Lemma. — Let K,L∈D(T,Λ) and assume that Extⁱ(K,L)=0 for i < 0.

Then U→HomD(U,Λ)(K|U,L|U) is a sheaf.

Proof. — Let H denote the complex Rhom(K,L). By assumption the natural map H →τ≥0H is an isomorphism. It follows that Hom_D(U,Λ)(K|U,L|U) is equal to the value of H⁰(H) on U which implies the lemma.

The existence part is more delicate. Let A denote the fibered and cofibered category over ∆ whose fiber over [n] ∈ ∆ is the category of Λ-modules in Un. For a morphism α: [n] → [m], F∈A(n) and G∈A(m) we have

Hom_α(F,G)=Hom_A(m)(α^∗F,G)=Hom_A(n)(F, α∗G).

We write A⁺ for the restriction of A to ∆⁺. Define a new category tot(A⁺) as follows:

• The objects of tot(A⁺) are collections of objects (Aⁿ)ⁿ≥0 with Aⁿ ∈A(n).

• For two objects (Aⁿ) and (Bⁿ) we define Hom_tot(A⁺₎((Aⁿ), (Bⁿ)):=

α

Hom_α(A^s(α),B^b(α)), where the product is taken over all morphisms in ∆⁺.

• If f = (f_α) ∈ Hom((Aⁿ), (Bⁿ)) and g = (g_α) ∈ Hom((Bⁿ), (Cⁿ)) are two morphisms then the composite is defined to be the collection of morphisms whose α component is defined to be

(g◦f)α :=

α=βγ

g_βf_γ

where the sum is taken over all factorizations of α (note that this sum is finite).

The category tot(A⁺) is an additive category.

Let (K,d) be a complex in tot(A⁺) so for every degree n we are given a family of objects (Kⁿ)^m ∈A(m). Set

K^n,m :=(K^n+m)ⁿ.

For α : [n] → [m] in ∆⁺ let d(α) denote the α-component of d so d(α)∈Hom_α((K^p)ⁿ, (K^p+1)^m)=Hom_α(K^n,p−n,K^m,p+1−m)

or equivalently d(α) is a map K^n,p → K^m,p+n−m+1. In particular, d(id_[n]) defines a map K^n,m → K^n,m+1 and as explained in [2, 3.2.8], this map makes K^n,∗

a complex. Furthermore for any α the map d(α) defines an α-map of complexes K^n,∗ → K^m,∗ of degree n−m+1. The collection of complexes K^n,∗ can also be defined as follows. For an integer p let L^pK denote the subcomplex with (L^pK)^n,m equal to 0 if n < p and K^n,m otherwise. Note that for any α : [n] → [m] which is not the identity map [n] → [n] the image of d(α) is contained in L^p+1K. Taking the associated graded of L we see that

grⁿ_LK[n] =(K^n,∗,d)

where d denote the differential (−1)ⁿd(id_[n]). Note that the functor (K,d)→K^n,∗

commutes with the formation of cones and with shifting of degrees.

As explained in [2, 3.2.8], a complex in tot(A⁺) is completely character- ized by the data of a complex K^n,∗ ∈ C(A⁺) for every [n] ∈ ∆⁺ and for every morphism α : [n] → [m] an α-morphism d(α) :K^n,∗ → K^m,∗ of degree n−m+1, such that d(id_[n]) is equal to (−1)ⁿ times the differential of K^n,∗ and such that for every α we have

α=βγ

d(β)d(γ)=0.

Via this dictionary, a morphism f :K→ L in C(tot(A⁺)) is given by an α-map f(α):K^n,∗→K^m,∗ of degree n−m for every morphism α: [n] → [m] in ∆⁺ such that for any morphism α we have

α=βγ

d(β)f(γ)=

α=βγ

f(β)d(γ).

Let K(tot(A⁺)) denote the category whose objects are complexes in tot(A⁺) and whose morphisms are homotopy classes of morphisms of complexes. The cat- egory K(tot(A⁺)) is a triangulated category. Let L⊂K(tot(A⁺)) denote the full subcategory of objects K for which each K^n,∗ is acyclic for all n. The category L

is a localizing subcategory of K(tot(A⁺)) in the sense of [3, 1.3], and hence the localized category D(tot(A⁺)) exists. The category D(tot(A⁺)) is obtained from K(tot(A⁺)) by inverting quasi-isomorphisms. Recall that an object K∈K(tot(A⁺)) is called L-local if for any object X ∈ L we have HomK(tot(A⁺))(X,K) = 0. Note that the functor K→K^n,∗ descends to a functor

D(tot(A⁺))→D(Un,Λ).

We define D⁺(tot(A⁺))⊂D(tot(A⁺)) to be the full subcategory of objects K for which there exists an integer N such that H^j(K^n,∗)=0 for all n and all j≤N.

Recall [3, 4.3], that a localization for an object K∈K(tot(A⁺)) is a morph- ism K→I with I an L-local object such that for any L-local object Z the natural map

HomK(tot(A⁺))(I,Z)→HomK(tot(A⁺))(K,Z) (2.3.ii)

is an isomorphism.

2.3.5. Lemma. — A morphism K→I is a localization if I is L-local and for every n the map K^n,∗→I^n,∗ is a quasi-isomorphism.

Proof. — By [3, 2.9], the morphism (2.3.ii) can be identified with the natural map

Hom_D(tot(A⁺₎₎(I,Z)→Hom_D(tot(A⁺₎₎(K,Z), (2.3.iii)

which is an isomorphism if K→ I induces an isomorphism in D(tot(A⁺)). 2.3.6. Proposition. — Let K∈C(tot(A⁺)) be an object with each K^n,∗ homotopically injective. Then K is L-local.

Proof. — Let X ∈ L be an object. We have to show that any morphism f :X→K in C(tot(A⁺)) is homotopic to zero. Such a homotopy h is given by a collection of maps h(α) such that

f(α)= −

α=βγ

d(β)h(γ)+h(β)d(γ).

We usually write just h for h(id_[n]).

We construct these maps h(α) by induction on b(α)−s(α). For s(α) =b(α) we choose the h(α) to be any homotopies between the maps f(id_[n]) and the zero maps.

For the inductive step, it suffices to show that Ψ(α)=f(α)+d(α)h+hd(α)+

α=βγ

d(β)h(γ)+h(β)d(γ)

commutes with the differentials d, where

α=βγ denotes the sum over all possible factorizations with β and γ not equal to the identity maps. For then Ψ(α) is homotopic to zero and we can take h(α) to be a homotopy between Ψ(α) and 0.

Define

A(α)=

α=βγ

d(β)h(γ)+h(γ)d(β)

and

B(α)=d(α)h+hd(α)+A(α).

2.3.7. Lemma. — One has the identity

α=βγ

A(β)d(γ)−d(β)A(γ)=

α=βγ

h(β)S(γ)−S(β)h(γ),

where S(α) denotes

α=βγd(β)d(γ).

Proof

α=βγ

A(β)d(γ)−d(β)A(γ)

=

α=ργ

d()h(ρ)d(γ)+h()d(ρ)d(γ)−d()h(ρ)d(γ)−d()d(ρ)h(γ)

=

α=βγ

h(β)S(γ)−S(β)h(γ),

where

α=ργ denotes the sum over all possible factorizations with , ρ, and γ

not equal to the identity maps.

2.3.8. Lemma. — One has the identity

α=βγ

B(β)d(γ)−d(β)B(γ)

= −h(d(α)d +dd(α))+(d(α)d+dd(α))h+

α=βγ

h(β)S(γ)−S(β)h(γ).

Proof

α=βγ

B(β)d(γ)−d(β)B(γ)

=

α=βγ

d(β)hd(γ)+hd(β)d(γ)+A(β)d(γ)−d(β)d(γ)h

−d(β)hd(γ)−d(β)A(γ)

= −h(d(α)d+dd(α))+(d(α)d+dd(α))h+

α=βγ

h(β)S(γ)−S(β)h(γ).

We can now prove 2.3.6. We compute

dA(α)−A(α)d

=

α=βγ

dd(β)h(γ)+dh(β)d(γ)−d(β)h(γ)d −h(β)d(γ)d

=

α=βγ

dd(β)h(γ)+(−f(β)−B(β)−h(β)d)d(γ)

−d(β)(−f(γ)−B(γ)−dh(γ))−h(β)d(γ)d

=

α=βγ

dd(β)h(γ)−f(β)d(γ)−B(β)d(γ)

−h(β)dd(γ)+d(β)f(γ)+d(β)B(γ)+d(β)dh(γ)−h(β)d(γ)d

=

α=βγ

(−S(β)h(γ))+h(β)S(γ)−f(β)d(γ)+d(β)f(γ) +h(d(α)d+dd(α))−(d(α)d+dd(α))h−

α=βγ

h(β)S(γ)−S(β)h(γ)

= f(α)d −df(α)+fd(α)−d(α)f +h(d(α)d+dd(α))

−(d(α)d +dd(α))h. So finally

dΨ(α)−Ψ(α)d

=df(α)+dd(α)h+dhd(α)+dA(α)−f(α)d −d(α)hd−hd(α)d

−A(α)d

=df(α)+dd(α)h+dhd(α)−f(α)d−d(α)hd−hd(α)d +f(α)d

−df(α)+fd(α)−d(α)f +hd(α)d +hdd(α)−d(α)dh−dd(α)h

=0.

This completes the proof of 2.3.6.

Let

^∗:C(A(∅))→C(tot(A⁺))

be the functor sending a complex K to the object of C(tot(A⁺)) with ^∗K^n,∗ = K|Un with maps d(id_[n]) equal to (−1)ⁿ times the differential, for ∂i : [n] → [n+1] the map d(∂i) is the canonical map of complexes, and all other d(α)’s are zero.

The functor ^∗ takes quasi-isomorphisms to quasi-isomorphisms and hence induces a functor

^∗:D(A(∅))→D(tot(A⁺)).

2.3.9. Lemma. — The functor ^∗ has a right adjoint R∗ : D(tot(A⁺)) → D(A(∅)) and R∗ is a triangulated functor.

Proof. — We apply the adjoint functor theorem [22, 4.1]. By our assumptions the category D(A(∅)) is compactly generated. Therefore it suffices to show that ^∗ commutes with coproducts (direct sums) which is immediate.

More concretely, the functor R∗ can be computed as follows. If K is L-local and there exists an integer N such that for every n we have K^n,m =0 for m < N, then R∗K is represented by the complex with

(∗K)^p=

n+m=p

n∗K^n,m with differential given by

d(α). This follows from Yoneda’s lemma and the ob- servation that for any F∈D(A(∅)) we have

HomD(A(∅))(F,R∗K)=HomD(tot(A⁺))(^∗F,K)

=HomK(tot(A⁺))(^∗F,K) since K is L-local

=Hom_K(A(∅))(F, ∗K) by [2, 3.2.12]. 2.3.10. Lemma. — For any F∈D⁺(A(∅)) the natural map F→R∗^∗F is an isomorphism.

Proof. — Represent F by a complex of injectives. Then ^∗F is L-local by 2.3.6.

The result then follows from cohomological descent.

Let D⁺_cart(tot(A⁺) ⊂ D⁺(tot(A⁺)) denote the full subcategory of objects K such that for every n and inclusion ∂i : [n]_{→ [}n+1] the map of complexes

K^n,∗|U_n+1 →K^n+1,∗

is a quasi-isomorphism.

2.3.11. Proposition. — Let K∈D⁺_cart(tot(A⁺)) be an object. Then ^∗R∗K→K is an isomorphism. In particular, R∗ and ^∗ induce an equivalence of categories between D⁺_cart(tot(A⁺)) and D⁺(A(∅)).

Proof. — For any integer s and system (K^n,∗,d(α)) defining an object of C(tot(A⁺)) we obtain a new object by (τ≤sK^n,∗,d(α)) since for any α which is not the identity morphism the map d(α) has degree ≤ 0. We therefore obtain a functor τ≤s:C(tot(A⁺))→C(tot(A⁺)) which takes quasi-isomorphisms to quasi- isomorphisms and hence descends to a functor

τ≤s:D(tot(A⁺))→D(tot(A⁺)).

Furthermore, there is a natural morphism of functors τ≤s →τ≤s+1 and we have Khocolimτ≤sK.

Note that the functor ^∗ commutes with homotopy colimits since it commutes with direct sums. If we show the proposition for the τ≤sK then we see that the natural map

^∗(hocolimR∗τ≤sK)hocolim^∗R∗τ≤sK→ hocolimτ≤sKK is an isomorphism. In particular K is in the essential image of ^∗. Write K=^∗F.

Then by 2.3.10 R∗KF whence ^∗R∗K→K is an isomorphism.

It therefore suffices to prove the proposition for K bounded above. Con- sidering the distinguished triangles associated to the truncations τ≤sK we further reduce to the case when K is concentrated in just a single degree. In this case, K is obtained by pullback from an object of A(∅) and the proposition again

follows from 2.3.10.

For an object K∈K(tot(A⁺)), we define τ≥sK to be the cone of the natural map τ≤s−1K→K.

Observe that the category K(tot(A⁺)) has products and therefore also homo- topy limits. Let K∈K_C(tot(A⁺)) be an object. By 2.3.11, for each s we can find a bounded below complex of injectives Is ∈ C(A(∅)) and a quasi-isomorphism σs : τ≥sK → ^∗Is. Since ^∗Is is L-local and ^∗ : D⁺(A(∅)) → D(tot(A⁺)) is fully faithful by 2.3.11, the maps τ≥s−1K → τ≥sK induce a unique morphism ts :Is−1→ Is in K(A(∅)) such that the diagrams

τ≥s−1K −−−→ τ≥sK

σs−1⏐⏐ ⏐⏐^σs

^∗Is−1 t_s

−−−→ ^∗Is

commutes in K(tot(A⁺)).

2.3.12. Proposition. — The natural map K→holim^∗Is is a quasi-isomorphism.